Re: Fourier transform - Question

Unit step function if and , otherwise.
First line is the Fourier transform (FT) of step function.
Second line is the FT of a generic function which is shifted by .
Third line is irrelevant for current derivation.

First, the given function can be illustrated as difference of two unit step functions. Note that if and , otherwise. Therefore, if and , otherwise. Multiply it by , you get the desired function.

Using the FT results given in the top, you can write the FT of and its shifted version resulting,

Note that the delta function is defined as for and , i.e. only when , . Thus, it is clear that is 0 when .
However, at , . Therefore, the product of and becomes 0 for even .
Conclusion is that is always 0.